Title: Some non-trivial elements of higher homotopy groups of the space of long n-knots

Speaker: Leo Yoshioka (University of Tokyo)

Abstract:  We give a class of cycles in the space of long n-knots R^n --> R^{n+2} that are constructed from specific two or three-loop graphs. Then, we show that invariants given through configuration space integrals detect the non-triviality of these cycles. As a corollary, we give another proof of the non-finite generation of homotopy groups of the space of long n-knots, which is shown by Budney, Gabai and Watanabe. Interestingly, our construction also gives non-trivial cycles of the space of long n-knots R^n --> R^{n+k} with higher codimension k >  2. 

We modify configuration space integrals to address obstructions called hidden faces, so that the integrals give well-defined invariants. In this modification, we introduce a new graph-complex for configuration space integrals. We show that this complex is quasi-isomorphic to the graph-complex which Arone and Turchin introduced in the context of embedding calculus.